Stable cohomotopy in codimensions 2 and 3 receives complete algebraic characterizations for CW complexes and bordism interpretations for manifolds, yielding necessary and sufficient conditions for nowhere-vanishing vector bundle sections.
Taylor,The principal fibration sequence and the second cohomotopy set, Proceedings of the Freedman Fest, Geom
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Stable Cohomotopy in Codimensions Two and Three: From Algebraic Characterizations to Bordism-Theoretic Interpretations
Stable cohomotopy in codimensions 2 and 3 receives complete algebraic characterizations for CW complexes and bordism interpretations for manifolds, yielding necessary and sufficient conditions for nowhere-vanishing vector bundle sections.